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On some iterated means arising in homogenization theory. (English) Zbl 1099.26002

The authors consider iterations of arithmetic and power means (in particular the arithmetic, geometric and harmonic means) and discuss methods for determining their limits. In particular, they study the iterated means \((A_mP_m)^ m\) and \((P_mA_m)^ m\), in which \(A_m(x)=\alpha x+\beta\) and \(P_m(x)=(\gamma x^ {1/1-p}+\delta )^ {1-p}\), where \(\gamma =v^ {1/mn}\), \(\alpha =v^{ n-1/mn}\) and \(\gamma +\delta =\alpha +\beta =1\). It turns out that these iterated means converge to the same limit, which is in general difficult to find, but in the case \(p=2\) it is known to be equal to \(1+nv(x-1)/(n+(1-v)(x-1))\). This result was proved in the PhD thesis of the first author by obtaining explicit formulae for the iterated means. In the paper under review an alternative (matrix) proof of these formulae is given. Moreover, the case \(n=1\) is treated in detail, and some results are established for the general case and the limiting cases.

MSC:

26A18 Iteration of real functions in one variable
35M20 PDE of composite type (MSC2000)
26D99 Inequalities in real analysis

Keywords:

iteration
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References:

[1] J. Arazy, T. Claesson, S. Janson and J. Peetre: Means and their iterations. In: Proceedings of the Nineteenth Nordic Congress of Mathematics, Reykjavik. 1984, pp. 191-212. · Zbl 0606.26007
[2] M. Avellaneda: Iterated homogenization, differential effective medium theory and applications. Comm. Pure Appl. Math. 40 (1987), 527-554. · Zbl 0629.73010
[3] E. F. Beckenbach: Convexity properties of generalized mean value functions. Ann. Math. Statistics 13 (1942), 88-90. · Zbl 0061.11601
[4] A. Bensoussan, J. L. Lions, and G. C. Papanicolaou: Asymptotic Analysis for Periodic Structures. North Holland, Amsterdam-New York-Oxford, 1978. · Zbl 0404.35001
[5] J. Bergh, J. L?fstr?m: Interpolations Spaces. An introduction (Grundlehren der mathematischen Wissenschaften 223). Springer-Verlag, Berlin-Heidelberg-New York, 1976.
[6] W. E. Boyce, R. C. Diprima: Elementary Differential Equations and Boundary Value Problems. John Wiley & Sons, New York, 1986.
[7] A. Braides, D. Lukkassen: Reiterated homogenization of integral functionals. Math. Models Methods Appl. Sci. 10 (2000), 47-71. · Zbl 1010.49011
[8] D. A. G. Bruggerman: Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen. Ann. Physik. 24 (1935), 634.
[9] P. S. Bullen, D. S. Mitrinovi?, and P. M. Vasi?: Means and Their Inequalities. D. Reidel Publishing Company, Dordrecht, 1988.
[10] G. H. Hardy, J. E. Littlewood, and G. P?lya: Inequalities. Cambridge University Press, Cambridge, 1934 (1978).
[11] Z. Hashin, S. Shtrikman: A variational approach to the theory of effective magnetic permeability of multiphase materials. J. Appl. Phys. 33 (1962), 3125-3131. · Zbl 0111.41401
[12] J.-L. Lions, D. Lukkassen, L.-E. Persson, and P. Wall: Reiterated homogenization of monotone operators. C. R. Acad. Sci. Paris, S?r. I, Math. 330 (2000), 675-680. · Zbl 0953.35041
[13] J.-L. Lions, D. Lukkassen, L.-E. Persson, and P. Wall: Reiterated homogenization of nonlinear monotone operators. Chinese Ann. Math. Ser. B 22 (2001), 1-12. · Zbl 0979.35047
[14] D. Lukkassen: Formul? and bounds connected to homogenization and optimal design of partial differential operators and integral functionals. PhD thesis (ISBN: 82-90487?87-8). University of Troms?, 1996.
[15] D. Lukkassen: A new reiterated structure with optimal macroscopic behavior. SIAM J. Appl. Math. 59 (1999), 1825-1842. · Zbl 0933.35023
[16] J. Peetre: Generalizing the arithmetic-geometric mean?a hapless computer experiment. Internat. J. Math. Math. Sci. 12 (1989), 235-245. · Zbl 0707.26005
[17] J. Peetre: Some observations on algorithms of the Gauss-Borchardt type. Proc. of the Edinburgh Math. Soc. (2) 34 (1991), 415-431. · Zbl 0746.39006
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